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\author{A. K. W. }
\title{Some Exercises in Linear Algebra }% by Artin }
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%\date{2020 年 2 月 28 日}
\date{April 23, 2024}

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%This is a collection of exercises from the book by Artin. 
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%\section{Matrices}

\begin{enumerate}

\item  
Verify the associative law for the matrix product
$
\begin{bmatrix} 1&2 \\ 3&1 \end{bmatrix}
\begin{bmatrix}  1&3&2 \\ 2&1&3  \end{bmatrix}
\begin{bmatrix}  3 \\ 1 \\ 2  \end{bmatrix}
$.  


\item  
Compute the matrix products 
$\begin{bmatrix} 1&a \\ 0&1 \end{bmatrix}
\begin{bmatrix}  1&b \\ 0&1  \end{bmatrix}$
and $\begin{bmatrix}  1&a \\ 0&1  \end{bmatrix}^n$. 


\item  
Find a formula for $\begin{bmatrix} 1&1&1 \\ 0&1&1 \\ 0&0&1 \end{bmatrix}^n$, and prove it by induction. 

\item  
Let $D$ be the diagonal matrix with diagonal entries $d_1,\cdots, d_n$, and let $A = (a_{ij})$ be an arbitrary $n\times n$ matrix. 
Compute the products $DA$ and $AD$.

\item  
Prove that the product of upper triangular matrices is upper triangular.

\item  
Find all matrices that commute with the given matrix 
$\begin{bmatrix} 0&1 \\ 0&0 \end{bmatrix}$.

\item  
A square matrix $A$ is nilpotent if $A^k =0 $ for some $k>0$. 
Prove that if $A$ is nilpotent, then $I + A$ is invertible. 
Do this by finding the inverse.

\item  
Find all solutions of the equation $x_1 + x_2 + 2x_3 - x_4 = 3$.

\item  
Find inverses of the matrix 
$\begin{bmatrix} 3&6 \\ 1&8 \end{bmatrix}$. 

\item  
Prove that if a product $AB$ of $n\times n$ matrices is invertible, so are the factors $A$ and $B$.

\item  
Consider an arbitrary system of linear equations $AX = B$, where $A$ and $B$ are real matrices. Prove that if the system of equations $AX = B$ has more than one solution then it has infinitely many.

\item  
Let $A$ be a square matrix. 
Show that if the system $AX = B$ has a unique solution for some
particular column vector B, then it has a unique solution for all $B$.

\item  
A matrix $B$ is symmetric if $B = B^t$. 
Prove that for any square matrices $B$, $BB^t$ and $B + B^t$ are symmetric, and that if $A$ is invertible, then $(A^{-1})^t = (A^t)^{-1}$.

\item  
Let $A$ and $B$ be symmetric $n \times n$ matrices. Prove that the product $AB$ is symmetric if and only if $AB = BA$.

\item  
How much can a matrix be simplified if both row and column operations are allowed?

\item  
Let $A$ be an $n \times n$ matrix. Determine $\det(-A)$ in terms of $\det(A)$.

\item  
Let $A$ be an $n\times n$ matrix with integer entries $a_{ij}$. 
Prove that $A$ is invertible, and that its inverse $A^{-1}$ has integer entries, if and only if $\det(A)=\pm 1$.

\item  
Prove that the numbers of the form $a+bi$, where $a$ and $b$ are rational numbers, form a
subfield of $\mathbb{C}$.

\item  
Find the inverse of $5$ modulo $p$, for $p = 7, 11, 13$, and $17$.

\item  
Consider the system of linear equations 
$
\begin{bmatrix} 2&3 \\ -1&3 \end{bmatrix}
\begin{bmatrix} x_1 \\ x_2 \end{bmatrix}
=\begin{bmatrix} 6 \\ 5 \end{bmatrix}.
$
Solve the system in $\mathbb{F}_{11}$.

\item  
Find a basis for the vector space of $2\times 2$ symmetric matrices.

\item  
Let $A$ be an $m \times n$ matrix, and let $A'$ be the result of a sequence of elementary row
operations on $A$. Prove that the rows of $A$ span the same space as the rows of $A'$.

\item  
Let $V = F^n$ be the space of column vectors. 
Prove that every subspace $W$ of $V$ is the space of solutions of some system of homogeneous linear equations $AX = 0$.

\item  
Prove that the set $B = ((1, 1, 0)^t, (0, 1, 2)^t, (1, 3 ,2)^t)$ is a basis of $\mathbb{R}^3$. Find the coordinate vector of the vector $v = (3,2,3)^t$ with respect to this basis.

\item  
Determine the base change matrix in $\mathbb{R}^2$, when the old basis is the standard basis
$E = (e_1, e_2)$ and the new basis is $B = (e_1 + e_2, e_1 - e_2)$.

\item  
Let $A$ be an $\ell\times m$ matrix and let $B$ be an $n\times p$ matrix. 
Prove that the rule $M\mapsto AMB$ defines a linear transformation from the space $F^{m\times n}$ of $m\times n$ matrices to the space $F^{\ell\times p}$.

\item  
Let $v_1,\cdots,v_n$ be elements of a vector space $V$. 
Prove that the map $\varphi:F^n\to V$ defined by 
$\varphi(X) = v_1x_1+\cdots+v_nx_n$ is a linear transformation. 

\item  
Let $A$ be an $m\times n$ matrix. 
Use the dimension formula to prove that the space of solutions
of the linear system $AX = 0$ has dimension at least $n-m$.

\item  
Prove that every $m\times n$ matrix $A$ of rank 1 has the form $A = XY^t$, where $X,Y$ are $m-$ and $n-$dimensional column vectors. 
How uniquely determined are these vectors?

\item  
Let $A$ and $B$ be $2\times 2$ matrices. Determine the matrix of the operator $T:M\mapsto AMB$ on the space $F^{2\times 2}$ of $2\times 2$ matrices, with respect to the basis $(e_{11}, e_{12}, e_{21}, e_{22})$ of $F^{2\times 2}$.

\item  
Find all real $2\times 2$ matrices that carry the line $y = x$ to the line $y = 3x$.

\item  
Determine the dimensions of the kernel and the image of the linear operator $T$ on the space $\mathbb{R}^n$ defined by $T(x_1,\cdots,x_n)=(x_1+x_n,x_2+x_{n-1},\cdots,x_n+x_1)^t$.

\item  
Let $T:V \to V$ be a linear operator on a vector space of dimension 2. 
Assume that $T$ is not multiplication by a scalar. 
Prove that there is a vector $v$ in $V$ such that $(v,T(v))$ is a basis of $V$, and describe thc matrix of $T$ with respect to that basis.

\item  
Let $B$ be a complex $n\times n$ matrix. Prove or disprove: The linear operator $T$ on the space of all $n\times n$ matrices defined by $T(A)=AB-BA$ is singular.

\item  
Let $T$ be a linear operator on a vector space $V$, and let $\lambda$ be a scalar. The eigenspace $V^{(\lambda)}$ is the set of eigenvectors of $T$ with eigenvalue $\lambda$, together with 0. Prove that $V^{(\lambda)}$ is a $T$-invariant subspace.

\item  
Let $T$ be a linear operator on a vector space $V$. 
Prove that if $W_1$ and $W_2$ are $T$-invariant subspaces of $V$, then $W_1+W_2$ and $W_1\cap W_2$ are $T$-invariant.

\item  
A $2\times 2$ matrix $A$ has an eigenvector $v_1 =(1,1)^t$ with eigenvalue 2 and also an eigenvector $v_2 = (1,2)^t$ with eigenvalue 3. Determine $A$.

\item  
Let $P$ be the real vector space of polynomials $p(x) = a_0 + a_1x +\cdots + a_nx^n$ of degree at most $n$, and let $D$ denote the derivative $\frac{d}{dx}$, considered as a linear operator on $P$.
Prove that $D$ is a nilpotent operator, meaning that $D^k = 0$ for sufficiently large $k$.

\item  
Let $T$ be a linear operator on a finite-dimensional vector space for which every nonzero vector is an eigenvector. Prove that $T$ is multiplication by a scalar.

\item  
The characteristic polynomial of the matrix 
$\begin{bmatrix} 0&1&2 \\ 1&1&0 \\ 1&*&* \end{bmatrix}$ 
is $t^3-4t-1$. Determine the missing entries.

\item  
Which real $2\times 2$ matrices have real eigenvalues? Prove that the eigenvalues are real if the off-diagonal entries have the same sign.

\item  
Let $V$ be a vector space with basis $(v_0,\cdots,v_n)$ and let $a_0,\cdots,a_n$ be scalars. Define a linear operator $T$ on $V$ by the rules $T(v_i) = v_{i+1}$ if $i < n$ and $T(v_n) = a_0v_0 + a_1v_1 +\cdots + a_nv_n$.
Determine the matrix of $T$ with respect to the given basis, and the characteristic polynomial of $T$.

\item  
Do $A$ and $A^t$ have the same eigenvectors? the same eigenvalues?

\item  
Consider the linear operator of left multiplication by an $m\times m$ matrix $A$ on the space $F^{m\times m}$ of all $m\times m$ matrices. Determine the trace and the determinant of this operator.

\item  
Let $A$ and $B$ be $n\times n$ matrices. Determine the trace and the determinant of the operator on the space $F^{n\times n}$ defined by $M\mapsto AMB$.

\item  
Suppose that a complex $n\times n$ matrix $A$ has distinct eigenvalues $\lambda_1, \cdots,\lambda_n$, and let $v_1,\cdots,v_n$ be eigenvectors with these eigenvalues. Show that every eigenvector is a multiple of one of the vectors $v_i$.

\item  
Let $T$ be a linear operator that has two linearly independent eigenvectors with the same eigenvalue $A$. Prove that $A$ is a multiple root of the characteristic polynomial of $T$.

\item  
Let $A= \begin{bmatrix} 2&1 \\ 1&2 \end{bmatrix}$. Find a matrix $P$ such that 
$P^{-1}AP$ is diagonal, and find a formula for the matrix $A^{30}$.

\item  
Suppose that $A$ is diagonalizable. Can the diagonalization be done with a matrix $P$ in the special linear group?
The special linear group is the group of real $n\times n$ matrices with determinant 1. 

\item  
Prove that if $A$ and $B$ are $n\times n$ matrices and $A$ is nonsingular, then $AB$ is similar to $BA$. Is this true when $A$ and $B$ are both singular?

\item  
A linear operator $T$ is nilpotent if some positive power $T^k$ is zero. 
Prove that $T$ is nilpotent if and only if there is a basis of $V$ such that the matrix of $T$ is upper triangular, with diagonal entries zero.

\item 
Find all real $2\times 2$ matrices such that $A^2 = I$, and describe geometrically the way they operate by left multiplication on $\mathbb{R}^2$.

\item  
Prove that $A =\begin{bmatrix} 1&1&1 \\ -1&-1&-1 \\ 1&1&1 \end{bmatrix}$ is an idempotent matrix, i.e., that $A^2 = A$, and find its Jordan form.

\item  
Let $V$ be a complex vector space of dimension 5, and let $T$ be a linear operator on $V$ whose characteristic polynomial is $(t-\lambda)^5$. Suppose that the rank of the operator $T-\lambda I$ is 2. What are the possible Jordan forms for $T$?

\item  
Determine the Jordan form of the matrix
$A=\begin{bmatrix} 1&1&0 \\ 0&1&0 \\ 0&1&1 \end{bmatrix}$.

\item  
Determine all possible Jordan forms for a matrix whose characteristic polynomial is $(t+2)^2(t-5)^3$.

\item  
Find a $2\times 2$ matrix with entries in $\mathbb{F}_p$ that has a power equal to the identity and an eigenvalue in $\mathbb{F}_p$, but is not diagonalizable.

\item  
Is every complex square matrix $A$ such that $A^2 = A$ diagonalizable?

\item  
Determine all invariant subspaces of a linear operator whose Jordan form consists of one block.

\item  
Let $A$ and $B$ be diagonalizable complex matrices. Prove that there is an invertible matrix $P$ such that $P^{-1}AP$ and $P^{-1}BP$ are both diagonal if and only if $AB = BA$.

\item  
Let $A$ be an $n\times n$ matrix. Prove the formula $\exp(\mathrm{tr}(A))=\det(\exp(A))$. 

\item 
Determine the matrices that represent the following rotations of $\mathbb{R}^3$:
\begin{enumerate}
\item angle $\theta$, the axis $e_2$, 
\item angle $2\pi/3$, axis contains the vector $(1,1,1)^t$, 
\item angle $\pi/2$, axis contains the vector $(1,1,0)^t$.
\end{enumerate}

\item 
Use row operation to put the matrix $\begin{bmatrix} 1&1&2&3 \\ 2&3&1&1 \\ 3&2&1&4 \end{bmatrix}$ into echelon form.

\item 
Calculate the inverse of the matrix $A=\begin{bmatrix} 1&0&1 \\ 1&1&0 \\ 0&1&1 \end{bmatrix}$ by applying row operations to the augmented matrix $(A\mid E)$, where $E$ is the identity matrix. 

\item 
Determine if the system of equations has a solution, and if so give the most general solution,
\begin{eqnarray*}
\left\{\begin{array}{lll}
x+2y+z &=& 2,\\ 
-x+2y &=& -1,\\ 
5x-2y+2z &=& 7.
\end{array}\right.
\end{eqnarray*}

\item 
Calculate the adjugate of the matrix $A=\begin{bmatrix} 2&-1&0 \\ 0&3&1 \\ -1&2&3 \end{bmatrix}$. 

\item 
Find a basis for the subspace $V=\{(x,y,z,t)^t\mid x+y+2t=0, y-3z=0\}$ of $\mathbb{R}^4$. 

\item 
Calculate the angle between the vector $(1,0,1)^t$ and the vector $(-1,1,0)^t$ in $\mathbb{R}^3$. 

\item 
Compute the projection of the vector $s=(5,2,1)^t$ in the direction of the vector $r=(-1,2,3)^t$. 

\item 
Let $\langle \vec{x},\vec{y}\rangle = 2x_1y_1+3x_2y_2+2x_3y_3+x_1y_3 +x_3y_1 -2x_2y_3 -2x_3y_2$, 
where $\vec{x}=(x_1,x_2,x_3)^t$ and $\vec{y}=(y_1,y_2,y_3)^t$ are vectors in $V=\mathbb{R}^3$. 
Determine whether $\langle-,- \rangle$ is an inner product of $V$. 
If it is an inner product, write down the appropriate form of the Cauchy-Schwarz inequality. 

\end{enumerate}

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